Theorem upfval 49663

Description: Function value of the class of universal properties. (Contributed by Zhi Wang, 24-Sep-2025.) (Proof shortened by Zhi Wang, 12-Nov-2025.)

Hypotheses

Ref	Expression
upfval.b	⊢ 𝐵 = (Base‘𝐷)
upfval.c	⊢ 𝐶 = (Base‘𝐸)
upfval.h	⊢ 𝐻 = (Hom ‘𝐷)
upfval.j	⊢ 𝐽 = (Hom ‘𝐸)
upfval.o	⊢ 𝑂 = (comp‘𝐸)

Assertion

Ref	Expression
upfval	⊢ (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})

Distinct variable groups:   𝐵,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝐶,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝐷,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐸,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐻,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐽,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝑂,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦

Proof of Theorem upfval

Dummy variables 𝑏 𝑐 𝑑 𝑒 ℎ 𝑗 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6849	. . . 4 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) ∈ V)
2		fveq2 6834	. . . . . 6 ⊢ (𝑑 = 𝐷 → (Base‘𝑑) = (Base‘𝐷))
3	2	adantr 480	. . . . 5 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) = (Base‘𝐷))
4		upfval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
5	3, 4	eqtr4di 2790	. . . 4 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) = 𝐵)
6		fvexd 6849	. . . . 5 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) ∈ V)
7		simplr 769	. . . . . . 7 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑒 = 𝐸)
8	7	fveq2d 6838	. . . . . 6 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = (Base‘𝐸))
9		upfval.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝐸)
10	8, 9	eqtr4di 2790	. . . . 5 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = 𝐶)
11		fvexd 6849	. . . . . 6 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) ∈ V)
12		simplll 775	. . . . . . . 8 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → 𝑑 = 𝐷)
13	12	fveq2d 6838	. . . . . . 7 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) = (Hom ‘𝐷))
14		upfval.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐷)
15	13, 14	eqtr4di 2790	. . . . . 6 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) = 𝐻)
16		fvexd 6849	. . . . . . 7 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) ∈ V)
17		simp-4r 784	. . . . . . . . 9 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → 𝑒 = 𝐸)
18	17	fveq2d 6838	. . . . . . . 8 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) = (Hom ‘𝐸))
19		upfval.j	. . . . . . . 8 ⊢ 𝐽 = (Hom ‘𝐸)
20	18, 19	eqtr4di 2790	. . . . . . 7 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) = 𝐽)
21		fvexd 6849	. . . . . . . 8 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) ∈ V)
22		simp-5r 786	. . . . . . . . . 10 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → 𝑒 = 𝐸)
23	22	fveq2d 6838	. . . . . . . . 9 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) = (comp‘𝐸))
24		upfval.o	. . . . . . . . 9 ⊢ 𝑂 = (comp‘𝐸)
25	23, 24	eqtr4di 2790	. . . . . . . 8 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) = 𝑂)
26		simp-6l 787	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑑 = 𝐷)
27		simp-6r 788	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑒 = 𝐸)
28	26, 27	oveq12d 7378	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑑 Func 𝑒) = (𝐷 Func 𝐸))
29		simp-4r 784	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑐 = 𝐶)
30		simp-5r 786	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑏 = 𝐵)
31	30	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑥 ∈ 𝑏 ↔ 𝑥 ∈ 𝐵))
32		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑗 = 𝐽)
33	32	oveqd 7377	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑤𝑗((1st ‘𝑓)‘𝑥)) = (𝑤𝐽((1st ‘𝑓)‘𝑥)))
34	33	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥)) ↔ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))))
35	31, 34	anbi12d 633	. . . . . . . . . . 11 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ↔ (𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥)))))
36	32	oveqd 7377	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑤𝑗((1st ‘𝑓)‘𝑦)) = (𝑤𝐽((1st ‘𝑓)‘𝑦)))
37		simplr 769	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → ℎ = 𝐻)
38	37	oveqdr 7388	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
39		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑜 = 𝑂)
40	39	oveqd 7377	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦)) = (⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦)))
41	40	oveqd 7377	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))
42	41	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ 𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
43	38, 42	reueqbidv 3379	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
44	36, 43	raleqbidv 3312	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
45	30, 44	raleqbidv 3312	. . . . . . . . . . 11 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
46	35, 45	anbi12d 633	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))))
47	46	opabbidv 5152	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))} = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})
48	28, 29, 47	mpoeq123dv 7435	. . . . . . . 8 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
49	21, 25, 48	csbied2 3875	. . . . . . 7 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → ⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
50	16, 20, 49	csbied2 3875	. . . . . 6 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → ⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
51	11, 15, 50	csbied2 3875	. . . . 5 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → ⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
52	6, 10, 51	csbied2 3875	. . . 4 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
53	1, 5, 52	csbied2 3875	. . 3 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
54		df-up 49661	. . 3 ⊢ UP = (𝑑 ∈ V, 𝑒 ∈ V ↦ ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}))
55		ovex 7393	. . . 4 ⊢ (𝐷 Func 𝐸) ∈ V
56	9	fvexi 6848	. . . 4 ⊢ 𝐶 ∈ V
57	55, 56	mpoex 8025	. . 3 ⊢ (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) ∈ V
58	53, 54, 57	ovmpoa 7515	. 2 ⊢ ((𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
59		reldmup 49662	. . . 4 ⊢ Rel dom UP
60	59	ovprc 7398	. . 3 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = ∅)
61		reldmfunc 49562	. . . . . 6 ⊢ Rel dom Func
62	61	ovprc 7398	. . . . 5 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 Func 𝐸) = ∅)
63	62	orcd 874	. . . 4 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → ((𝐷 Func 𝐸) = ∅ ∨ 𝐶 = ∅))
64		0mpo0 7443	. . . 4 ⊢ (((𝐷 Func 𝐸) = ∅ ∨ 𝐶 = ∅) → (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) = ∅)
65	63, 64	syl 17	. . 3 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) = ∅)
66	60, 65	eqtr4d 2775	. 2 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
67	58, 66	pm2.61i 182	1 ⊢ (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃!wreu 3341  Vcvv 3430  ⦋csb 3838  ∅c0 4274  ⟨cop 4574  {copab 5148  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  1^st c1st 7933  2^nd c2nd 7934  Basecbs 17170  Hom chom 17222  compcco 17223   Func cfunc 17812   UP cup 49660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-func 17816  df-up 49661

This theorem is referenced by:  upfval2  49664  uppropd  49668  reldmup2  49669  relup  49670  uprcl  49671